Deriving Entropic Geodesics from the Master Entropic Equation (MEE) of the Theory of Entropicity (ToE): Einstein's Relativity as a Weak Field Limit of ToE 

Entropic Geodesics can be derived from the Master Entropic Equation (MEE) in the Theory of Entropicity (ToE) via the Euler-Lagrange formalism applied to test particles in the entropy field $$S(x)$$, analogous to how GR geodesics emerge from the metric variation.

## Variational Principle

Start with the Obidi Action restricted to a test particle coupled to $$S(x)$$: $$S_{\text{test}} = \int \left[ -m \sqrt{-g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu} + \lambda(S) \nabla_\mu S \dot{x}^\mu \right] d\tau$$, where $$\lambda(S)$$ is a Lagrange multiplier enforcing entropic extremization, and dots denote $$d/d\tau$$. This encodes paths minimizing "entropic resistance" along MEE solutions.[8]

## Derivation Steps

1. Vary $$S_{\text{test}}$$ w.r.t. $$x^\mu(\tau)$$: The metric term yields the standard GR geodesic $$\frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta} \dot{x}^\alpha \dot{x}^\beta = 0$$.

2. The entropic coupling adds $$\frac{\delta}{\delta x^\mu} \int \lambda(S) \partial^\nu S \dot{x}_\nu d\tau = f^\mu$$, where $$f^\mu \propto g^{\mu\nu} \left( \partial_\nu S + S \Gamma^\lambda_{\nu\lambda} \right) / \chi(S)$$ from MEE's $$\square S + \frac{dV}{dS} = \eta T^\mu_\mu$$. 

3. Combined: $$\frac{D \dot{x}^\mu}{d\tau} = f^\mu(S, \nabla S)$$, with $$f^\mu$$ sourced by MEE entropy gradients, ensuring irreversibility.

## GR Limit

When $$\nabla S \to 0$$ (weak fields), $$f^\mu \to 0$$, recovering pure GR geodesics; otherwise, dissipation emerges from $$\partial S / \partial \tau > 0$$. [8][4] This derivation aligns ToE's unification, treating gravity as entropic optimization.

Citations:

[1] A derivation of the master equation from path entropy maximization - PMC https://pmc.ncbi.nlm.nih.gov/articles/PMC4108628/

[2] Entropy-Time Geodesics as a Universal Framework for ... https://arxiv.org/pdf/2512.20648.pdf

[3] A Brief Note on Some of the Beautiful Implications ... https://johnobidi.substack.com/p/a-brief-note-on-some-of-the-beautiful

[4] On the Conceptual and Mathematical Foundations of ... https://client.prod.orp.cambridge.org/engage/coe/article-details/68ea8b61bc2ac3a0e07a6f2c

[5] The Theory of Entropicity (ToE) Derives and Explains Mass ... https://client.prod.orp.cambridge.org/engage/coe/article-details/6900d89c113cc7cfff94ef3a

[6] Can Schrodinger equation be derived from universal ... https://www.facebook.com/groups/1685507044817357/posts/25162414503366614/

[7] The Theory of Entropicity (ToE) Derives and Explains Mass ... https://papers.ssrn.com/sol3/papers.cfm?abstract_id=5673430

[8] Scalar Entropic Field theory, or Entropy First https://www.reddit.com/r/HypotheticalPhysics/comments/1los2kh/here_is_a_hypothesis_scalar_entropic_field_theory/